{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:GFWTYTI5W35X4C3QCQUF6HCEKI","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"674940e61e47863bbe2b817e949a2dbe988c45b089e9587be773561fb8ea87c0","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GN","submitted_at":"2016-10-28T14:46:34Z","title_canon_sha256":"4c68d48e7c816f1d72d89ffb83cca7d4fadeb092b15d83165d8ca604bf474634"},"schema_version":"1.0","source":{"id":"1610.09245","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1610.09245","created_at":"2026-05-18T01:00:58Z"},{"alias_kind":"arxiv_version","alias_value":"1610.09245v1","created_at":"2026-05-18T01:00:58Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1610.09245","created_at":"2026-05-18T01:00:58Z"},{"alias_kind":"pith_short_12","alias_value":"GFWTYTI5W35X","created_at":"2026-05-18T12:30:19Z"},{"alias_kind":"pith_short_16","alias_value":"GFWTYTI5W35X4C3Q","created_at":"2026-05-18T12:30:19Z"},{"alias_kind":"pith_short_8","alias_value":"GFWTYTI5","created_at":"2026-05-18T12:30:19Z"}],"graph_snapshots":[{"event_id":"sha256:fe225131ee3ba0f416d64dcf48c90a06602848a5a2e82e1fad96aa204f8c5139","target":"graph","created_at":"2026-05-18T01:00:58Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"We introduce the cardinal invariant $aL^\\prime(X)$ and show that $|X|\\leq 2^{aL^\\prime(X)\\chi(X)}$ for any Hausdorff space $X$ (a corollary of Theorem 4.4. This invariant has the properties a) $aL^\\prime(X)=\\aleph_0$ if $X$ is H-closed, and b) $aL(X)\\leq aL^\\prime(X)\\leq aL_c(X)$. Theorem 4.4 then gives a new improvement of the well-known Hausdorff bound $2^{L(X)\\chi(X)}$ from which it follows that $|X|\\leq 2^{\\psi_c(X)}$ if $X$ is H-closed (Dow/Porter [5]). The invariant $aL^\\prime(X)$ is constructed using convergent open ultrafilters and an operator $c:\\scr{P}(X)\\to\\scr{P}(X)$ with the prope","authors_text":"Jack Porter, Nathan Carlson","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GN","submitted_at":"2016-10-28T14:46:34Z","title":"On the cardinality of Hausdorff spaces and H-closed spaces"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1610.09245","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:095d38787e8a20e8ef305f5b941e4b4232e1306d8dff9586ea0190d9566e21cc","target":"record","created_at":"2026-05-18T01:00:58Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"674940e61e47863bbe2b817e949a2dbe988c45b089e9587be773561fb8ea87c0","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GN","submitted_at":"2016-10-28T14:46:34Z","title_canon_sha256":"4c68d48e7c816f1d72d89ffb83cca7d4fadeb092b15d83165d8ca604bf474634"},"schema_version":"1.0","source":{"id":"1610.09245","kind":"arxiv","version":1}},"canonical_sha256":"316d3c4d1db6fb7e0b7014285f1c4452101ac9db20a4104edc713c28e736af5d","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"316d3c4d1db6fb7e0b7014285f1c4452101ac9db20a4104edc713c28e736af5d","first_computed_at":"2026-05-18T01:00:58.660686Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:00:58.660686Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"japOPk2tDz4gu4K7l8QNCV0CQxqct9GtOX9hGnClWNI2QtU02PDI3vezMAZl0g/HKxPu8gh4BHGPL/3drrr4Dw==","signature_status":"signed_v1","signed_at":"2026-05-18T01:00:58.661343Z","signed_message":"canonical_sha256_bytes"},"source_id":"1610.09245","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:095d38787e8a20e8ef305f5b941e4b4232e1306d8dff9586ea0190d9566e21cc","sha256:fe225131ee3ba0f416d64dcf48c90a06602848a5a2e82e1fad96aa204f8c5139"],"state_sha256":"6481b26e40ed3fdad5d92835711e2773ddd944a79caddb37f028962d6057271a"}